An algebraic cycle relating to a section with high penetration
Let \(X\) be a smooth complex projective variety and \(E\) a holomorphic vector bundle on \(X\). If \(E\) admits a flat hermitian structure then the mapping \(\omega^ p:H^ t(X,\Omega^ s_ X\otimes E)\to H^{t+p}(X,\Omega_ X^{s+p}\otimes E)\), determined by the class \(\omega\) of a hyperplane section, is an isomorphism. The author looks at the kernel of this mapping from a point of view avoiding an explicit mention of curvature. He considers the case \(t=s=0\) --- when the elements of \(H^ t\) are sections of \(E\). He shows that if a section is transformed to the zero section and not in the kernel, then there are consequences for the algebraic cycles, provided a weakened form of the Hodge conjecture holds. The main tool is Deligne's mixed Hodge theory.
scientific article; zbMATH DE number 8639
scientific article